Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 73034 by mathmax by abdo last updated on 05/Nov/19

calculate U_n =Σ_(k=1) ^n  (k/((k+1)!))

$${calculate}\:{U}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}}{\left({k}+\mathrm{1}\right)!} \\ $$

Commented by mathmax by abdo last updated on 06/Nov/19

we have U_n =Σ_(k=1) ^n  ((k+1−1)/((k+1)!)) =Σ_(k=1) ^n ((1/(k!))−(1/((k+1)!)))  =1−(1/(2!))+(1/(2!))−(1/(3!)) +....+(1/(n!))−(1/((n+1)!)) =1−(1/((n+1)!))

$${we}\:{have}\:{U}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}+\mathrm{1}−\mathrm{1}}{\left({k}+\mathrm{1}\right)!}\:=\sum_{{k}=\mathrm{1}} ^{{n}} \left(\frac{\mathrm{1}}{{k}!}−\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)!}\right) \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{2}!}−\frac{\mathrm{1}}{\mathrm{3}!}\:+....+\frac{\mathrm{1}}{{n}!}−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)!}\:=\mathrm{1}−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)!} \\ $$

Answered by mind is power last updated on 05/Nov/19

Un=Σ_(k=1) ^n (1/(k!))−(1/((k+1)!))=1−(1/((n+1)!))

$$\mathrm{Un}=\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}!}−\frac{\mathrm{1}}{\left(\mathrm{k}+\mathrm{1}\right)!}=\mathrm{1}−\frac{\mathrm{1}}{\left(\mathrm{n}+\mathrm{1}\right)!} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com